I think neither/nor is more suited as a pair of coordinating conjunctions for this type of construction:

"Neither Paul nor James likes her very much."
"James doesn't like her, nor does Paul."

Anyone please correct me if I'm wrong, but I think the purpose of the "not only... but also" construction is to present new information in the second part. "Not only this one thing happened, but also this other different thing too!" 

In Latin, this construction (*non solum*... *sed etiam*) appears a lot in Cicero who used it to add weight to his points.

It kind of breaks down though if you're saying the same thing both times:

"Not only does James not like her, but also Paul doesn't either."

You start having to use the auxiliary form of the verb to not repeat yourself, so it's not a true 1:1 construction where each of the clauses has its own unique verb.

It's okay. But neither/nor is more suited to the parallel idea of the woman not being liked, and doesn't create the expectation of you presenting new information.